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Abstract

This research work focuses on the estimation of barrier and lookback option prices using
finite difference numerical methods. Here, we aim at approximating the fair prices
of the zero rebate up-and-out and down-and-out knock out barrier options, as well as
the fixed strike lookback options. Simulation and finite difference techniques will be
used to approximate these prices. The Monte-Carlo simulation, the antithetic Monte-
Carlo simulations and the Crank-Nicolson approach will be specifically employed on
the barrier options. Other finite difference methods like the implicit and the explicit
method will be discussed but the Crank-Nicolson method will be employed in the
numerical valuations owing to its accuracy in comparison to others. Next, the fixed
strike lookback option prices will be estimated using the Monte-Carlo and the antithetic
Monte-Carlo simulation methods. An extended version of the Black-Scholes
model will be used in the valuation of their exact prices owing to their exotic nature.
The Monte-Carlo and the antithetic Monte-Carlo methods are next employed to simulate
the values of these option prices. The resulting prices will be compared to the
exact fair prices and this will be followed by some error analysis.
From the findings, the antithetic method gave the best option price estimate in comparison
to the ordinary Monte-Carlo method when the simulation approach was used.
It will also be observed that the Monte-Carlo simulation had a slow rate of convergence
as a result of higher variances of the estimate from the true solution. Hence,
such ineficiency was curbed by the introduction of antithetic Monte-Carlo simulation
which had smaller variances of the estimate, and this in turn gave a better estimate.
Furthermore, it will also be observed that the Crank-Nicolson method converged
faster with increase in the discretisation steps of the underlying asset and the time.